Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

1999 ◽  
Vol 66 (3) ◽  
pp. 742-749 ◽  
Author(s):  
S. M. Lin
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Equations ◽  
Excitation Frequency ◽  
Solution Procedure ◽  
Variable Coefficients ◽  
The Laplace Transform ◽  
Governing Differential Equation ◽  
Systematic Solution ◽  
Elastically Restrained

A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

scholarly journals New Bi-modular Material Approach to Buckling Problem of Reinforced Concrete Columns

2016 ◽  
Vol 6 (1) ◽  
pp. 19 ◽  
Author(s):  
Ahmad Salah Edeen Nassef ◽  
Mohammed A. Dahim
Keyword(s):  
Differential Equation ◽  
Reinforced Concrete ◽  
Differential Equations ◽  
Concrete Columns ◽  
Neutral Axis ◽  
Reinforced Concrete Columns ◽  
New Approach ◽  
Codes Of Practice ◽  
Governing Differential Equation ◽  
Transverse Deflection

<p class="1Body">This paper was investigating the buckling problem of reinforced concrete columns considering the reinforced concrete as bi – modular material. Governing differential equations was driven. The relation between the non-dimensional transverse deflection and non-dimensional distance between centroid axis and the neutral axis "eccentricity" was drawn to enable the solution of the governing differential equation. The new approach was verified with different experimental results and different codes of practice.<strong></strong></p>

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

Rotating Disk Shape: Is the Equation Nonlinear?

1989 ◽  
Vol 111 (4) ◽  
pp. 456-458
Author(s):  
R. R. Jettappa
Keyword(s):  
Differential Equation ◽  
Linear Equation ◽  
Rotating Disk ◽  
Variable Coefficients ◽  
Thin Disk ◽  
Second Order ◽  
First Order ◽  
Governing Differential Equation ◽  
Centrifugal Loading

The determination of the shape of a rotating disk under centrifugal loading is considered. It is shown that the governing differential equation for the shape of a rotating thin disk is reducible to a linear equation of second order with variable coefficients. However, the form of this equation is such that it can be treated as an equation of first order thereby facilitating the integration by quadratures. All this is possible without any additional mathematical assumptions so that the results are exact within the limitations of the thin disk theory.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

CONDITIONS OF EXISTENCE THE TRACE OF FUNCTIONS FROM SPACE WITH MULTIWEIGHTED DERIVATIVES IN A SPECIAL POINT

2020 ◽  
pp. 123-128
Author(s):  
А. Kalybay ◽  
◽  
Zh. Keulimzhaeva ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Functional Space ◽  
Variable Coefficients ◽  
Weight Functions ◽  
Weight Sobolev Space ◽  
Strongly Degenerate

When solving differential equations with variable coefficients, especially when the coefficients degenerate at the boundary of a given domain, problems arise in the formulation of boundary value problems. Usually, differential equations with variable coefficients are investigated in a suitable weight functional space. Often in the role of such spaces the weight Sobolev space or various generalizations are considered, which are currently sufficiently studied. However, in some cases, when the coefficients of the considered differential equation are strongly degenerate, the formulation of boundary value problems becomes problematic. In this work, we consider the so-called space with multiweighted derivatives, where after each derivative, the function is multiplied by the weight function and then the next derivative is taken. By controlling the behavior of the weight functions on the boundary, strongly degenerate equations can be investigated. Here we investigate the existence of traces on the boundary of a function from such spaces.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

2020 ◽  
Vol 8 (3) ◽  
pp. 61-68
Author(s):  
Avyt Asanov ◽  
Kanykei Asanova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Linear Differential Equations ◽  
Common Solution ◽  
The Common ◽  
Linear Differential ◽  
New Formula

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Shao-Wen Yao
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type ◽  
The Laplace Transform ◽  
First Time

Purpose This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution. Design/methodology/approach First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed. Findings A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He’s polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly. Originality/value The proposed approach is original and very effective because this approach is, to the authors’ knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest.

scholarly journals Nonlocal Symmetries for Time-Dependent Order Differential Equations

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 771
Author(s):  
Andrei Ludu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Harmonic Numbers ◽  
Nonlocal Symmetries ◽  
The Riemann Zeta Function ◽  
Symmetry Of The Solution ◽  
New Type ◽  
Riemann Zeta

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

scholarly journals Some further results on oscillation of neutral differential equations

1992 ◽  
Vol 46 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Jianshe Yu ◽  
Zhicheng Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Image Position

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficientswhere P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

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